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Cisse, I. and Bardan, Gérald and Mojtabi, Abdelkader
Rayleigh Bénard convective instability of a fluid under
high-frequency vibration. (2004) International Journal of Heat and
Mass Transfer, 47 (19-20). 4101-4112. ISSN 0017-9310
Official URL:
https://
doi.org/10.1016/j.ijheatmasstransfer.2004.05.002
Rayleigh B
enard convective instability of a fluid under
high-frequency vibration
I. Cisse, G. Bardan
*, A. Mojtabi
UPS-IMFT, UFR MIG, Departement de Mecanique, 118 route de Narbonne, 31000 Toulouse, France
Abstract
The generation of two-dimensional thermal convection induced simultaneously by gravity and high-frequency vibration in a bounded rectangular enclosure or in a layer is investigated theoretically and numerically. The horizontal walls of the container are maintained at constant temperatures while the vertical boundaries are thermally insulated, impermeable and adiabatic. General equations for the description of the time-averaged convective flow and, within this framework, the generalized Boussinesq approximation are formulated. These equations are solved using a spectral collocation method to study the influence of vibrations (angle and intensity). Hence, a theoretical study shows that mechanical quasi-equilibrium (i.e., state in which the averaged velocity is zero but the oscillatory component is in general non-zero) is impossible when the direction of vibration is not parallel to the temperature gradient. In the other case, it is proved that the mechanical equilibrium is linearly stable up to a critical value of the unique stability parameter, which depends on the vibrational field. In this paper, it is shown that high-frequency vertical oscillations can delay convective instabilities and, in this way, reduce the convective flow. The isotherms are oriented perpendicular to the axis of vibration. In the case where the direction of vibration is perpendicular to the temperature gradient, small values of the Grashof number, the stability parameter, induce the generation of an average convective flow. When the aspect ratio is large enough, the character of the bifurcation is practically the same as in the limiting case of an infinitely long layer.
Keywords: Vibration; Rayleigh Benard
1. Introduction
Vibrations are known to be among the most effective ways of affecting the behaviour of fluid systems, in the sense of increasing or reducing the convective heat transfer. Most of the material in this paper is devoted to the case of high-frequency vibration (i.e., when the period of vibration is much smaller than the reference hydrodynamic times and the displacement amplitude is smaller than the height of the cell). We focus our
attention on the case where a container filled with a fluid heated from below is subjected to arbitrary high-frequency vibrations. The description of the thermo-vibrational flows in the limiting case of high-frequency and small amplitude of vibrations may be effectively obtained in the frame of the averaging method, which leads to the system of equations for averaged fields of velocity, pressure and temperature. Simonenko et al. [1,2], were the first to use this method and propose the time-averaged form of the Boussinesq equations. Sub-sequently, it was demonstrated that high-frequency vibrations are most relevant in modifying stability characteristics. Gershuni and Zhukhovitskii [3] intro-duced the vibrational analogue of the Rayleigh Number, Rav, to represent the intensity of the vibrational source. *
Corresponding author. Tel.: 61-55-6788; fax: +33-5-61-55-8326.
E-mail address:bardan@imft.fr(G. Bardan).
Thus, for an infinite fluid layer in weightlessness (i.e., when the static gravity is absent), only the specific thermovibrational mechanism is responsible for insta-bility excitation and quasi-equilibrium is possible. They studied the effects of vibration angle and the interaction between natural and thermovibrational convection. The configuration corresponding to a horizontal layer heated from below or from above under longitudinal vibra-tion is considered by Braverman and Oron [4] and by Gershuni and Zhukhovitskii [3]. They show that the state of quasi-equilibrium becomes unstable when the vibrational Rayleigh number exceeds some critical value which depends on the boundary conditions at the par-allel planes bounding the fluid layer. At a given a (angle between axis of vibration and isothermal surfaces), the neutral curve RavðkÞ assumes a minimum at some kc which determines the wave number of the most dan-gerous perturbation. The corresponding value Ravc
yields the equilibrium stability border. The numerical calculation implies that the stability is minimal for lon-gitudinal vibrations for Ravc¼ 2129 and kc¼ 3:23. The
critical value of the vibrational Rayleigh number in-creases monotonically with the angle of inclination of the vibration axis. When a!p
2, they found Ravc! 1
and kc! 0. In the presence of static gravity, the onset of convection is caused by both thermovibrational and thermogravitational mechanisms. Several examples involving both mechanisms were presented by Gershuni and Zhukhovitskii [5] and later by Braverman and Oron [4]. The experimental results given by Zavarykin et al. [6] are in good agreement with the theoretical ones. Thermovibrational convection in an enclosure has long been investigated during the last decade due to its extensive applications in engineering, like solar energy systems, electronic cooling equipment or crystal growth processes. There are many practical problems of natural convection in an enclosure which are caused by non-periodic (accelerating–decelerating) or non-periodic (har-monic vibration) motion. In the past, Richardson [7] reviewed the effects of sound and wall vibration on heat transfer. Forbes et al. [8] conducted experiments to investigate the enhancement of thermal convection heat transfer in a liquid-filled rectangular enclosure by vibration. The results showed that the vibration fre-quency and acceleration were the dominant factors that affected heat transfer.
Using the time averaged method, Gershuni and Zhukhovitskii [5] studied the vibrational thermal con-vection under weightlessness in a rectangular, cylindrical enclosure and a heated cylinder in an unconfined fluid. Due to the high frequency assumption, many important phenomena, like the resonant state and the detailed variation of the heat transfer rate could not be investi-gated. However, Yurkov [9] directly solved the Bous-sinesq-approximated governing equations to study the thermal convection induced by finite-frequency
vibra-tion under weightless condivibra-tions. From the results of the average Nusselt number, the parametric resonant phenomenon was found. A more exhaustive study was carried out by Fu and Shieh [10] to investigate the effects of the vibration frequency and Rayleigh number on the thermal convection in the enclosure. The vibration fre-quency was varied from 1 to 104 and three different values of the Rayleigh number were considered. According to the results, thermal convection can be di-vided into five regions: (i) quasi-static convection, (ii) vibration convection, (iii) resonant vibration convection, (iv) intermediate convection and (v) high-frequency vibration convection. When the Rayleigh number is large enough, (Ra¼ 106), gravitational thermal convec-tion dominates, and the vibraconvec-tion moconvec-tion does not markedly enhance the heat transfer rate. In contrast, in the low Rayleigh number (Ra¼ 104) case, except in the quasi-static convection region, the vibration thermal convection is dominant, and the vibration enhances the heat transfer significantly. So, Gershuni et al. [11], in studying the structures of the flows, report that the transition from the basic to a multicellular flow sets in at a critical value of the Rav. When the aspect ratio equals 8 or more, the character of the bifurcation is practically the same as in the limiting case of an infinitely long layer. The latter statement was confirmed by Khallouf et al. [12], Lizee [13], Bardan et al. [14] for a binary mixture when they studied the non-linear regimes of two-dimensional thermovibrational convection in rect-angular cavities subjected to a longitudinal temperature gradient and transversal axis of vibration. It was shown that the transition from a four-vortex regime to inver-sional symmetry took place at some critical value of Rav. This value increased monotonously as the aspect ratio AL increased. Thus, when AL 4, the limiting case of rest with conductive heat transfer occurred.
All these previous results were of great help for understanding the g-jitter effects during fluid and mate-rial science microgravity experiments. On available space platforms, a systematic characterization of the accelerations has shown that the microgravity environ-ment is dynamic, depending on many sources, e.g., aerodynamic forces, on-board machinery, crew opera-tions or servicing activities. It is recognized that the presence of g-disturbances may cause strong discrepan-cies, and so, the fluid science processes may be sub-stantially changed by g-disturbances. To reduce the convective contributions of these g-jitters, it is conve-nient to consider acceleration fields as an expansion of harmonic oscillations, and to use a time-averaged method formulation. Previously, there have been many studies on thermal convection in a gravitationally modulated fluid layer with rigid, isothermal boundaries heated from below or from above, see Gresho and Sani [15], Rosenblat and Tanaka [16]. Biringen and Dana-basoglu [17] studied the effects of gravity modulation in
a thermally driven rectangular enclosure for terrestrial and microgravity environments. The results showed that the destabilizing and stabilizing effects of gravity mod-ulation agreed with the theories of Gresho and Sani [15]. More recently, Monti and Savino [18,19] performed a numerical simulation of non-linear problems of TVC (thermovibrational convection) with reference to typical fluid science experiments on the space station. They established that the flow field is in general 3D, because of end wall effects and the heat exchange with the sur-roundings through the lateral walls (not perfectly insu-lated). In the case of g-jitter, we have to deal with multifrequency vibrations spanning the interval from 101to 103, and here too, a time averaged formulation is necessary in order to reduce the computation time. Savino et al. [20] have also studied a three-dimensional TVC in a closed cell filled with liquid, with an initial linear temperature and a g-jitter orthogonal to the density gradient. They revealed that the time-averaged convective motion due to thermovibrational effects arises because of non-linear coupling between density and acceleration oscillations. The 3D numerical results presented in their work in terms of velocity and tem-perature fields are helpful for the design of microgravity experiments and to evaluate the range of validity of the two-dimensional assumption invoked in previous cal-culations.
Our study concerns the effects of vibration in a finite rectangular box filled with fluid. Our aim is to under-stand the effects of g-jitter on this problem. We also deal with an unbounded layer filled with fluid subjected to a horizontal or vertical vibrational field. We restrict our work to the case of a pure fluid but we add the influence of arbitrary angles of vibration. In studying the equi-librium conditions, we have found general equations for the quasi-equilibrium and equilibrium states. According to the boundary conditions applied to different geome-tries, it is possible to know if quasi-equilibrium or equilibrium exists. Many of our stability results are concerned with a vibrational parameter which depends only on the vibrational effects (i.e., it does not depend on temperature difference). In comparison with previous works which deal with the vibrational Grashof number, the use of this specific parameter permits a better understanding of the vibrational effects on convective flows. Numerical computations have been performed, too, in order to foresee the theoretical results.
2. Problem description and basic equations
We consider two-dimensional thermovibrational convection in a container of height H and length L. Fig. 1 represents the flow configuration and coordinates system. The flow domain is ðx; zÞ 2 X ¼ ½0; L ½0; H . All the physical properties are taken to be constant. The
horizontal walls at z¼ 0 and z ¼ H are kept at constant and uniform temperatures h1 and h2, respectively. The situation where h1>h2is adopted. The vertical walls (at x¼ 0 and x ¼ L) are insulated. All the boundaries are assumed rigid. The fluid cavity with its boundaries is subjected to linear harmonic oscillations.
2.1. The mathematical model
The fluid in the cavity is considered to be Newtonian and to satisfy the Boussinesq approximation. The thermophysical properties are constant except for the density in the buoyancy term which depends linearly on the local temperature. The equation of state has the form (1).
qðhÞ ¼ qrefð b1 hðh hrefÞÞ ð1Þ where qref¼ qðhrefÞ is the density at standard tempera-ture href¼ h2 and bh¼q1ref
oq
oh the thermal expansion coefficient. By introducing the appropriate coordinates connected with oscillating systems, the gravitational field is replaced by the sum of the gravitational and the vibrational acceleration (2) in the momentum equation. g! g bx2sinðxtÞ k ð2Þ where k¼ cos ax þ sin az is the unit vector along the axis of vibration and a¼ ðx; kÞ is the angle of vibration, bis the displacement amplitude and x the angular fre-quency.
These hypotheses lead to the following dimensionless conservation equations for mass (3), momentum (4) and energy (5), with the Boussinesq approximation. Using the velocity u, the pressure p and the temperature h as independent variables, the non-dimensional equations of the Boussinesq model are
r u ¼ 0 ð3Þ
Fig. 1. Geometrical configuration and axis of coordinates. Sketch of the cavity configuration.
ou otþ ðu rÞu ¼ rp þ r 2 uþ Gr h z þ Grvh ^xsinð ^xtÞ k ð4Þ oh otþ ðu rÞh ¼ 1 Prr 2 h ð5Þ
In the above equations, using the viscous diffusion time as the time-scale factor, lengths are non-dimen-sionalized by H , velocity by m
H (m is the kinematic diffu-sivity), time byH2
m, x by m
H2and temperature relative to h2
by h1 h2. In line with the problem description, the corresponding non-dimensional forms of the boundary conditions are (6)–(9).
u¼ 0 onoX ð6Þ
h¼ 1 for z¼ 0; 8x ð7Þ
h¼ 0 for z¼ 1; 8x ð8Þ
oh
ox¼ 0 for x¼ 0 and AL;8z ð9Þ The problem includes five non-dimensional numbers; the Grashof number Gr¼gbhðh1h2ÞH3
m2 , the modified
vibrational Grashof number Grv¼bxbhðh1h2ÞH
m , the pul-sation ^x¼ xH2
m, the Prandtl number Pr¼ m
a(a is the heat diffusivity) and the aspect ratio AL¼L
H. The modified vibrational Grashof number is obtained by substituting g by bx2 in the Grashof number and rescaling by the ratio between the viscous diffusive time and the vibra-tional timeH2=m
1=x. In the limit of high-frequency and low amplitude, the effect of vibration is then determined by the product bx which appears in the definition of the modified vibrational Grashof number Grv.
2.2. The averaged flow equations
In the asymptotic case of high-frequency oscillations where the period of the displacement s¼2p
x is very low compared to the characteristic times of thermal and kinematic diffusion (sH2
a and s H2
m) the application of the averaging method of Simonenko [2] only allows the mean flow and mean temperature to be solved (see [21]).
Let us introduce the additional variable W which forms the solenoidal part of the temperature field con-tribution to the gravitational force and rn its conser-vative part in the following Helmholtz decompositions (10) and (11).
T k¼ W þ rn ð10Þ
r W ¼ 0 ð11Þ
The mean motion is then determined by (12)–(16).
r U ¼ 0 ð12Þ oU ot þ ðU rÞU ¼ rP þ r 2 Uþ GrT z þ1 2Gr 2 vW r T kð WÞ ð13Þ oT otþ ðU rÞT ¼ 1 Prr 2 T ð14Þ rT ^ k ¼ r ^ W ð15Þ r:W ¼ 0 ð16Þ
The boundary conditions are in accordance with the physical statement of the problem being (6)–(9) and (17):
W n ¼ 0 onoX ð17Þ
where n is the outside normal vector.
3. Linear stability
3.1. Mechanical equilibrium
An equilibrium solution in presence of an oscillating force is possible only under certain conditions of heat input and cavity shape. In this case, the time averaged body force is compensated by the pressure gradient. The question is whether the state of mechanical quasi-equi-librium (i.e., the state at which the mean velocity is zero, but the pulsational component is not in general) exists or not in our situation. In order to prove the existence of a pure conductive solution of the systems (12)–(16), we substitute U¼ Uxxþ Uzz¼ 0 ando
ot¼ 0, the equilibrium fields T¼ T0, W¼ W0 and P¼ P0 in this system and seek a valid solution for arbitrary values of the non-dimensional parameters. The following system (18)–(21) is obtained: rP0þ GrT0zþ1 2Gr 2 vW0 r T0ð k W0Þ ¼ 0 ð18Þ DT0¼ 0 ð19Þ rT0^ k ¼ r ^ W0 ð20Þ r W0¼ 0 ð21Þ
with the boundary conditions (22)–(25):
T0¼ 1 for z ¼ 0; 8x ð22Þ T0¼ 0 for z ¼ 1; 8x ð23Þ oT0
ox ¼ 0 for x ¼ 0 and AL;8z ð24Þ
W0 n ¼ 0 on oX: ð25Þ
Eq. (19) with boundary conditions (22) and (23) lead to T0¼ 1 z. Applying the curl to Eq. (18), we obtain the vibrational hydrostatic condition (26):
The equilibrium follows from (19)–(21) and (26) when supplemented by the corresponding boundary condi-tions (specification of temperature and the requirement that W0 n vanishes on the cavity walls). Using the sys-tem of partial differential Eqs. (20)–(26), we can find the solenoidal vector W0 as indicated below:
W0¼ W0xxþ W0zz where ð27Þ W0x¼ 1
2sinð2aÞ x cos að þ z sin aÞ c1xsin a
þ c1zcos aþ c ð28Þ
W0z¼ cos2ðaÞ x cos að þ z sin aÞ
þ c1xcos aþ c1zsin aþ c0 ð29Þ c1, c, c0are integration constants. We specify that, in this general solution W0, the boundary conditions (25) are not taken into account.
3.1.1. Horizontal layer
Let us consider the problem of mechanical quasi-equilibrium stability for an infinite plane layer bounded by two parallel rigid plates z¼ 0 and 1. Taking boundary conditions (25) into account, the values of W0x and W0z(28), (29) lead to the following quasi-equilibrium solution W0 z ¼ 0 for z¼ 0 and z ¼ 1: ð30Þ T0¼ 1 z ð31Þ and ox0x oz ¼ cos a ð32Þ
In this case, the solenoidal field is longitudinal. We suppose that the net flux of this field is equal zero, i.e. Rz¼1
z¼0W0 xdz ¼ 0 which leads to:
T0¼ 1 z ð33Þ W0z¼ 0 ð34Þ W0x¼ 1 2 z cos a ð35Þ
This solution for W0 has already been found by Gershuni and Lyubimov [21]. Therefore, we can con-clude that quasi-equilibrium exists for an infinite plane layer subject to arbitrary angles of vibration.
3.1.2. Rectangular cavity
This situation is markedly different from that men-tioned above. For arbitrary values of a (a6¼p
2), the quasi-equilibrium solution or the equilibrium solution are not possible and the thermovibrational convective flow sets in at infinitely small values of temperature difference. In fact, if boundary conditions (25) are taken into account in Eqs. (28) and (29), we have an
equilib-rium solution only for a¼p
2. This equilibrium state under vertical vibration exists independently of the Grashof and vibrational Grashof number. Conse-quently, we will deal with arbitrary angles of vibration for an unbounded layer but only with vertical vibration for a container. In both cases, quasi-equilibrium or equilibrium solutions exist.
3.2. Stability of the equilibrium solution In the following we write Gv¼1
2Gr 2
vand refer to it as the vibrational Grashof number.
It is convenient to rewrite mean field Eqs. (12)–(16) as evolution equations for two-dimensional perturba-tions about this equilibrium state. We denote these perturbations by (U0; P0; T0; W0) and introduce the fol-lowing streamfunction representations
Ux0 ¼ ow 0 oz and U 0 z ¼ ow0 ox ð36Þ Wx0¼ o/ 0 oz and W 0 z ¼ o/0 ox ð37Þ
As a result a positive streamfunction corresponds to a clockwise cell i.e. U0¼ curlðw0yÞ.
3.3. Horizontal layer with arbitrary angle of vibration Considering the mean field Eqs. (12)–(16), replacing the expressions U0
x, Uz0, Wx0, Wz0 defined in (36) and (37), and taking the curl of Eq. (13), the resulting equations are o ot Dw0 T0 0 0 B @ 1 C A ¼ D2 Gr o ox Gv o2 ox2 o ox D Pr 0 0 o ox D 0 B B B B B B @ 1 C C C C C C A w0 T0 /0 0 B @ 1 C A þ N1ðw0;w0Þ N3ðw0; T0Þ 0 0 B @ 1 C A þ Gv N1ð/0/0Þ N2ð/0T0Þ 0 0 0 B @ 1 C A þ Gv 1 2 z cos a o ox D/ 0oT0 ox 0 0 0 B B B @ 1 C C C A ð38Þ where for all pairsðf ; gÞ of real functions
N1ðf ; f Þ ¼of oz o3f oxoz2 þo 3f ox3 of ox o3f ox2oz þo 3f oz3 ð39Þ
N2ðf ; gÞ ¼ o 2f oxoz og oxþ of oz o2g ox2 o2f ox2 og oz of ox o2g oxoz ð40Þ N3ðf ; gÞ ¼of oz og ox of ox og oz ð41Þ
The associated boundary conditions are ow0 oz ! z¼0;1 ¼ w 0 z¼0;1¼ 0 ð42Þ ðT0Þ z¼0;1¼ 0 ð43Þ ð/0Þ z¼0;1¼ 0 ð44Þ 3.3.1. Analytical results
The resulting linear problem is solved by means of a Galerkin method using the following expansions w0¼X N i¼1 aiðz 1Þ 2 ziþ1eIkxeIXht ð45Þ T0¼X N i¼1 biðz 1ÞzieIkxeIXht ð46Þ /0¼X N i¼1 ciðz 1ÞzieIkxeIXht ð47Þ
where Xhand k are real numbers, I¼pffiffiffiffiffiffiffi1and i2 IN . For an unbounded layer, the truncation used (N ¼ 4) predicts that Rac¼ PrGrc¼ 1708:549 and kc¼ 3:116. This result is in good agreement with the result (Rac¼ PrGrc¼ 1707:762) found by Reid and Harris [22]. Fig. 2
presents the linear stability results for vibration at angle a. In Fig. 2(a) we recover the results of Gershuni and Lyubimov [21]. The critical point on the Ra¼ PrGr axis is the solution of the classic Rayleigh–Benard problem. This critical Rayleigh number Rac¼ 1708 is connected with the most dangerous perturbation of the wave-number kc¼ 3:116. This point on the PrGv axis corre-sponds to the case of pure weightlessness. In the range of inclination angles, the stability curves intersect the PrGv axis and go to negative Ra. For Gershuni and Lyubimov [21], this means that instability occurs when a system is heated from above. In order to understand what can happen on earth, we propose a new formulation which will be adapted with an experiment. In an experiment on earth we can start in the static case Gv¼ 0, increase the parietal temperature gradient DT and reach the critical Rayleigh number at which convection occurs. Then, for a fixed value of PrGv, we do the same and increase DT from 0 to its critical value corresponding to the onset of convection. For a sufficiently large fixed value of PrGv, the convection seems to occur at the beginning of the experimentation i.e. for Ra¼ 0. As Ra ¼ Prgbhðh1h2ÞH3
m2
this means on earth that DT¼ ðh1 h2Þ ¼ 0 and it fol-lows by definition that Gv¼ 0. This is in contradiction with the fact that we can always impose a value of PrGv and then increase DT . The problem comes from the formulation as both Ra and PrGv depend on DT . By setting Gv¼ G
v:Gr2with Gv¼12 b2x2m2
g2H4, the new
adimen-sional vibrational parameter G
v no longer depends on DT. Fig. 2(b) presents the same stability borders but in the planeðPrGr; PrG
vÞ. This formulation is then adapted to an experimentation on earth. Moreover, a particular
Fig. 2. (a) Stability borders in the plane (Ra, Pr2G
v) for different inclination angles of the vibration axis (see [21]). (b) Stability borders
in the plane (Ra, Pr2G
v) for different inclination angles of the vibration axis. In the limiting case of high intensity of the vibrational field
i.e. bx2! 1 the system will have a behaviour closer and closer to the one it would have under zero-gravity conditions (except for
a¼p
angle of vibration is clearly shown. As one can see on Fig. 2(b), the stability borders for vertical vibrations (a¼p
2) are markedly different from the others. For 0 6 a 6 ac where ac¼27p
100, Rac decreases with respect to G
v and Rac! 0 when Gv! 1. For ac6a < p
2, Racfirst increases then decreases with respect to G
v but finally Rac! 0 when G
v! 1. For a ¼ p
2, Rac increases with respect to G
v and Rac! 1 when Gv! 1. This differ-ent behaviour could be linked with the vertical sidewall effects in the presence of vibration. We have seen that mechanical quasi-equilibrium exists only for vertical vibration in a rectangular cavity, whereas it always exists for unbounded layers. This shows that, with vibration, an unbounded layer cannot exactly represent the physics of a very large rectangular cavity. In the following, we will keep the formulation in Gv as it is now classic (see the bibliography) in the studies of thermovibrational convection.
3.3.2. Stationary or oscillatory threshold
In the static case, the linear stability problem is determined by the product PrGr which defines the Rayleigh number Ra¼ PrGr. Moreover, the linearized Eq. (38) are self-conjugated and the principle of stabil-ity exchange applies i.e. only steady state bifurcations occur.
With vibration, the linearized Eq. (38) are not self-conjugated and we cannot demonstrate that the bifur-cations are always stationary. Nevertheless, for all the following results we have verified that no Hopf bifur-cation occurs and Xh is found to be a pure imaginary complex in each case we have solved. We should men-tion that the associated Grashof number and also Xh strongly depend on the Prandtl number and the product PrGr is not relevantfor the oscillatory regime. To con-clude, with vibration, only steady state bifurcations occur in the range of parameters (Pr, Gv) which we use in the following. In the framework of stationary bifurca-tions, the linear stability problem is, as in the static case, determined by Ra¼ PrGr and Rav¼ PrGrv. Within this formulation in Rayleigh number Ra, the Prandtl number Pr does not appear explicitly in the linear stability analysis except for Hopf bifurcation but we have seen there are none.
Nevertheless, we keep the formulation in terms of Grashof number Gr because it will be shown in the weakly non-linear analysis that the formulation in Rayleigh number is not sufficient to explicitly eliminate the Prandtl number.
3.4. Rectangular cavity and vertical vibration
For a cavity, the perturbation equations are also given by (38) but a must be set to a¼p
2.
At the boundaries of the cavity the streamfunctions vanish and thus
ow0 ox ! x¼0;AL ¼ ow 0 oz ! z¼0;1 ¼ w 0 oX¼ 0 ð48Þ Tz¼0;10 ¼ oT 0 ox x¼0;AL ¼ 0 ð49Þ ð/0Þ oX¼ 0 ð50Þ
In this case, we use the following expansions w0¼X
N
i¼1 XM
j¼1
aijðx ALÞ2xiþ1ðz 1Þ2zjþ1eIXht ð51Þ
T0¼X M j¼1 b0jðz 1ÞzjeIXhtþX N i¼1 XM j¼1 bij x iþ 2 AL iþ 1 xiþ1ðz 1ÞzjeIXht ð52Þ /0¼X N i¼1 XM j¼1 cijðx ALÞxiðz 1ÞzjeIXht ð53Þ
where Xhis a real number, I¼pffiffiffiffiffiffiffi1andði; jÞ 2 IN2. Equation (38) with boundary conditions (48)–(50) are invariant under two reflections. These operations are described by the operators Sx, Sz and So¼ SxSz defined by Sx w0 T0 w01 0 @ 1 A x; zð Þ ¼ w0 T0 w0 1 0 @ 1 A Að L x; zÞ ð54Þ Sz w0 T0 w01 0 @ 1 A x; zð Þ ¼ w0 T0 w0 1 0 @ 1 A x; 1ð zÞ ð55Þ So w0 T0 w01 0 @ 1 A x; zð Þ ¼ w0 T0 w01 0 @ 1 A Að L x; 1 zÞ ð56Þ The different reflections Sx, So, Szverify the relations S2
x ¼ Id, Sz2¼ Id and So2¼ Id. The representation C of the group Z2 Z2 is C fId; Sx; Sz; Sog, where Id is the identity operator and this group plays an important role in the bifurcation analysis described below. In the presence of this group, the conduction state can lose stability and bifurcate to one specific solution which possesses either So-symmetric, Sx-symmetric, Sz-sym-metric or has all the symmetry properties. The resulting bifurcations are pitchforks except for the last case. Without vibration, it is known that Sz-symmetric solu-tions are always unstable and result from at least a second primary bifurcation. The first two primary bifurcations are supercritical pitchforks with either So-symmetric or symmetric solutions [23]. The Sx-symmetric eigenmodes contain an even number of rolls in the x-direction, whereas the Sz-symmetric ones con-tain an even number of rolls in the z-direction. The
So-symmetric eigenmodes contain an odd number of cells in both directions. The eigenmodes which possess all the symmetry properties contain an even number of cells in both directions. For the first primary bifurcation, the eigenvector could be either Sx-symmetric or So -symmetric. Moreover, for second primary bifurcation, the eigenvector could be either Sz-symmetric or have all the symmetry properties (So, Sxand Sz) but this solution could not be observed as it remains unstable almost without vibration.
MAPLE and MATLAB were used to find the values of Rac, Xhc, aij, bijand cijat which a bifurcation occurs. The choice of the truncation used (N¼ 6 and M ¼ 6) for a square cavity is linked with the results of Mizu-shima [23] who studied the static case. When the rect-angle is long, it is convenient to take more test functions in theðOxÞ direction in order to get a better convergence of the solution. For example, for AL¼ 5, we took N¼ 11 and M ¼ 6 (see Tables 1 and 2). The numerical computations confirm this choice. As shown in Fig. 3, our results are in good agreement with those of Mizu-shima [23].
Before presenting the influence of a vibrational field, it is worthwhile to recall the following well-known re-sults concerning the static case. As previously men-tioned, the conductive solution can lose stability to four different states that have different symmetry properties. The neutral curves of modes which have the same symmetry do not intersect. This repulsion of the eigen-values was discussed by the stability of paths of sym-metry-breaking bifurcations by Crawford and Knobloch [24]. In fact, the mode which occurs (first primary bifurcation) is either So- or Sx-symmetric. The corre-sponding two neutral curves are plotted in Fig. 3. The aim of this study is to understand how the vertical vibrations will affect these neutral curves. For different values of PrGv, the graphs of the two first primary bifurcation points are shown in Fig. 4. The values of ALi
are given in Table 3.
We note, in Fig. 4, the influence of vertical vibrations on the critical Grashof number. Indeed, the value of critical Grashof number increases with the intensity of vibrations. The conductive solution stays steady for greater temperature differences. The vibrations act both on the stability threshold and on the nature of the flow. We note effectively that the differencejALiþ1 ALij rises
with the intensity of vibrations. This distance corre-sponds to the wavelength of the flow in an infinite layer. The stability thresholds translated up and dilated horizontally. Consequently, when the vibrations are not present and considering an aspect ratio of 5, we
Table 1
Convergence with increasing N and M
N M 2 2 3 2 3 3 4 4 5 5 6 6
Rac1 2804.1 2798.0 2645.7 2586.9 2586.1 2585.0
Rac2 7806.0 7806.0 7672.4 6752.0 6750.4 6742.5
Critical Rayleigh number Racof the neutral modes So(Rac1) and Sx(Rac2) vs. the couple N M for Pr ¼ 1, Gv¼ 0 and AL¼ 1.
Fig. 3. Critical Rayleigh number Racof the neutral modes So
and Sx vs. the aspect ratio AL. Comparison with the results of
Mizushima and also with an unbounded layer in the static case. For AL< AL1 So-symmetric one-cell flow, for ALi< AL< ALiþ1
ði þ 1Þ-cell flow which is So- symmetric for i odd or Sx
-sym-metric for i even.
Table 2
Convergence with increasing N and M
N M 2 2 3 2 3 3 4 4 5 5 6 6 8 6 10 6 11 6
Rac1 4532.2 2729.1 2733.5 2663.3 2126.3 1861.8 1804.2 1780.3 1778.2
Rac2 14602.5 7556.6 4527.3 4144.7 2208.0 2122.0 2021.3 1920.7 1917.8
note a convective flow with five rolls. For Gv¼ 200, AL4
stays below five and the onset of convective flow still begins with five rolls. Unlike when the intensity of vibration is increasing (Gv¼ 800), the horizontal
trans-Fig. 5. Bifurcation diagram in the Nu–Gr plane for Pr¼ 1,
AL¼ 2 and vertical vibration. Along the two branches plotted
for Gv¼ 0 and 2000, the flow is bicellular, whereas for Gv¼
5000 the flow pattern is monocellular.
Fig. 4. Critical Rayleigh number Racof the neutral modes Soand Sxvs. the aspect ratio AL. We also add the results concerning an
unbounded layer. The vertical vibrations increase the value of the critical Rayleigh number.
Table 3
Critical aspect ratios ALi where the two modes So- and Sx
-symmetric exchange for PrGv¼ 0, PrGv¼ 200 and PrGv¼ 800
PrGv 0 200 800
AL1 1.64 1.65 1.70
AL2 2.68 2.76 2.86
AL3 3.72 3.80 4.01
Fig. 6. Stability curves of critical Grashof number Grcvs. the
aspect ratio AL for Gv¼ 0, 800, 2000, 4000, 6000. Each
bifur-cation point is determined on the base of the numerical results obtained for the finite amplitude regimes. We note the influence of vertical vibrations as predicted by theoretical results. Inset: Influence of vibrations on distancesjALiþ1 ALij. Numbers ALi
are relative to the intersection points of the symmetry modes So
et Sx. These distances increase with the intensity of vibrations
and are related to the vibrational Grashof number by the linear evolution law:jALiþ1 ALij ¼ 0:0001 Gvþ 1:1:
lation on the right of AL4 due to vibrations is
suffi-cient for AL4 to be greater than five. So, we can
con-clude that the setting off of the convective motion is characterized by a four-roll regime. These results are confirmed by the numerical results (Figs. 5 and 6). Each point in Fig. 6 was obtained by plotting the complete bifurcation diagram as in Fig. 5 and noting the value of Rac.
4. Computations of finite amplitude regimes 4.1. Numerical method
The system of Eqs. (12)–(16) was solved numeri-cally using a spectral method. This is based on the projection–diffusion algorithm developed for solving the 2D–3D unsteady incompressible equations [25]. Its interest is to circumvent the difficulty posed by the gradient pressure at the borders of the cavity. Temporal integration consists of a semi-implicit second-order finite difference approximation. The linear (viscous) terms are treated implicitly using a second-order back-ward Euler scheme, while a second-order Adams– Bashforth scheme is employed for the nonlinear (advective) parts. When applied to an advection–diffu-sion equation such as
of
dtþ U rf ¼ aDf ð57Þ
the method leads to 3 2f nþ1 2fnþ1 2f n1 dt ¼ aDf nþ1 2ðU rf Þn ðU rf Þn1 ð58Þ This equation can be written in the following form of the Helmholtz equation
ðD hÞfnþ1¼ s ð59Þ
where h¼ 3
2adtis the Helmholtz constant and s is a scalar quantity containing all the terms known at time tn¼ ndt (n is the time level and dt is the time step). The temporal integration, therefore, transforms the systems into a Helmholtz problem arising from Eq. (14) coupled to the Poisson problems (15), (16) with appropriate boundary conditions. Eqs. (12) and (13) are transformed into a generalized Stokes problem and solved by the projec-tion–diffusion method of Khallouf [26]. All the sub-problems obtained are either Helmholtz or Poisson-like operators. A spectral method, namely one utilizing Legendre collocation points, is used in the spatial dis-cretization of the Helmholtz and Poisson-like operators. Successive diagonalization is implemented to invert these operators. We mention that the Stokes and Darcy– Euler solvers are direct and guarantee an accurate
spectral solution with divergence-free solenoidal fields over the entire domain, including the boundaries. 4.2. Vertical vibration
Fig. 5 represents the Nusselt vs. the Grashof number for Pr¼ 1 and AL¼ 2. As Gv increases, the stability threshold increases too. Hence, vertical vibrations have a stabilizing effect on convective flow. As the first pri-mary bifurcations are supercritical pitchforks with either So-symmetric or Sx-symmetric solutions, a weakly non-linear analysis arround the bifurcation point shows that Nu¼ KpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðGr GrcÞ
where K is a constant which depends on Gv [14]. For each value of Gv, as Nu2¼ KðGr GrcÞ, a simply extrapolation technique allows us to determine an accurate value of Grc. We use the finite amplitude results to plot the Fig. 6.
Fig. 6 computed for Gv¼ 0, 800, 2000, 4000 and 6000 represents the analogue of Fig. 4 which summarizes the theoretical results. The numerical results confirm all the theoretical ones. With the vibrations, the curves are translated up and dilated horizontally. This means that the vertical vibrations delay the onset of the con-vection and have a stabilizing effect on the convective flow. We can better understand the dilating effect due to vibrations by looking at the inset in Fig. 6. This shows that the critical distance jALiþ1 ALij is a linear
function of the vibrational Grashof number and, con-sequently, the distance increases with the intensity of vibrations.
4.3. Imperfect bifurcation
We deal now with an inclination of the axis of vibration from the vertical (Fig. 7). Computations were performed for a rectangular cavity (AL¼ 2) for (Gv¼ 0, 2000, 5000, 10,000 and 20,000). Our previous prediction, given by the linear theory in which there is no conduc-tive solution when the axis of vibration diverges from the temperature gradient, is confirmed here. Indeed, for a¼p
2 1
100and when the intensity of vibrations increases (Gv¼ 2000), we can observe quasi-conductive motion or weak convection in the cavity. The deviation of the stability curve drawn for (Gv¼ 2000, 5000, 10,000, 20,000) from the stability curve (Gv¼ 0) proves that there is no equilibrium solution for a6¼p
2. This is an imperfect bifurcation scenario. When the vibrational Grashof number increases markedly and attains a criti-cal vibrational Grashof number, the criticriti-cal Grashof number begins to diminish in comparison with the pre-vious results. This shows that, at a certain value of intensity of vibration, the vibrations have a destabilizing effect on the convection.
4.4. Horizontal vibration
The results for horizontal vibrations (a¼ 0) have been summarized in (Fig. 8) for Pr¼ 1 and AL¼ 2. This
kind of vibration is very effective in driving thermovib-rational convection. This is the case of non-equilibrium. When Gvincreases for example for (Gv¼ 2000), a small temperature difference applied to the horizontal walls can set off a weak convection movement. The quasi-conductive regime generated is, therefore, destabilized for a critical Grashof number below the stability threshold of the conductive regime (Gv¼ 0). The critical Grashof number continues to decrease with increasing vibrational Grashof number (inset Fig. 8). In micro-gravity, a convective regime is present in the cavity and increases with the intensity of vibrations.
5. Conclusion
The influence of vibrations of low amplitude and high-frequency on fluid subjected to a temperature dif-ference has been studied. In order to better understand these effects of vibration in both intensity and direction, we have described a theoretical and numerical study of thermovibrational convection in an infinite layer, in a square, and in a rectangular cavity, notably in the par-ticular case permitting the existence of an equilibrium or quasi-equilibrium state. We noted close agreement be-tween the theoretical and the numerical results. In presence of vibrations, an infinite layer cannot perfectly represent the physical phenomena observed in very long rectangular cavities. In fact, the wall contribution is significant because it implies that mechanical equilib-rium is impossible when the axis of vibrations is not along the temperature gradient. When a6¼p
2, the vibra-tions generate a convective flow for temperature differ-ences below those necessary for the generation of convection in the classical Rayleigh–Benard problem. As these vibrations are sufficiently intense, it is possible to set off convection in a microgravity environment (space stations). This environment is dynamic and re-veals three-dimensionnal accelerations known as g-jitters having frequencies varying between 101and 103 Hz, which can perturb the realization of experiments like the measurement of the Soret coefficient or the making of pure cristals. On the other hand, when the axis of vibration is vertical, vibrations can preserve the con-ductive regime above the classical threshold of Ray-leigh–Benard. We can conclude that it is possible to generate or suppress the convective regime created by thermal gradient by means of mechanical vibrations. Thermal control with the use of vibrations is then pos-sible.
Acknowledgement
We thank G. Couegnas for his work and his contri-bution to the present study.
Fig. 7. Influence of a small deviation from the special case of vertical vibration. Bifurcation diagram in the Nu–Gr plane for Pr¼ 1, AL¼ 2 and a ¼p2100p.
Fig. 8. Bifurcation diagram in the Nu–Gr plane for Pr¼ 1,
AL¼ 2 and horizontal vibration. Along all the branches plotted
for Gv¼ 0, 2000, 3000, 3700, 5000 the flow is bicellular. The
next figure displays an enlargement around the quasi-diffusive
regime (Nu¼ 1). Inset: enlargement around the quasi-diffusive
regime (Nu¼ 1) for horizontal vibration. Along all the branches
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